import numpy as np
from scipy.optimize import fsolve
def implicit_heun_time_numpy(f, y, dt):
"""
Approximates the solution of the differential equation
dy/dt = f(t, y) using Implicit Heun's method.
"""
predictor = y + dt * f(t, y)
# Define the implicit function to be solved using fsolve
implicit_func = lambda y_next: y + (dt / 2) * (f(t, y) + f(t + dt, y_next)) - y_next
# Solve the implicit function to get the corrector step using fsolve
-?> y_corrected = fsolve(implicit_func, predictor)[0]
return y_corrected
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def heun_step(state,rhs,dt,etol=0.000001,maxiters = 100):
'''Update a state to the next time increment
using the implicit Heun's method.'''
error = 1
next_state = state + rhs(state)*dt
################### New iterative correction ###############
for n in range(0,maxiters):
next_state_old = next_state
next_state = state + (rhs(state)+rhs(next_state))/2*dt
error = np.sum(np.abs((next_state-next_state_old)/next_state)
if error < etol:
break
############### end of iterative correction ###############
return next_state
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